| BTC (N = 2,081) | |
|---|---|
| Mean (sd) | 2,080; 0.22 ± 3.94 |
| median (Q1, Q3) | 2,080; 0.19 (-1.21, 1.78) |
| min | -37.16954 |
| max | 25.24717 |
| Cumulative Return | 466.6306 |
| Normally Distributed VaR | -6.25452002497649 |
| Non-Normally Distributed VaR | -6.05178991592173 |
The above statistics table is based on the percentage returns for the daily price valuation of bitcoin; it presents the mean, median, min, and max returns to demonstrate the range as well as the average return over the six year period from 2014 to 2020.
In regards to the mean return (%) of bitcoin, it can be noted that it is positive and relatively close in value to the median price meaning that over a daily basis, bitcoin is expected to provide approximately .22% return of an investment per day during a 6 year period. This can fluctuate depending on the valuation trend as the price of bitcoin can over a larger period be trending downwards, upwards or sideways. Given that the mean and the median are relatively close in value, there is limited skewness in the data, and the skewness is favoring higher returns since the mean is to the right of the median.
Colloquially, a bull market is defined as a financial securitity with a sustained increase in valuation; thus, a bear market is defined as a financial securitity with a sustained decrease in valuation; and a trendless market does not have either a sustained increase or a sustained decrease in valuation.
The standard deviation is approximately 3.94% which means that 95% of the daily returns will most likely fluctuate between [-7.66%, 8.01%]. Anything that occurs outside of this interval would be a rare event with a probability \(\le 5%\), which means that most of the returns would occur be centered around the mean, and 5% of the returns would be \([-\infty,-7.66\%]\cup[8.01\%,\infty]\) (outside of the interval [-7.66%, 8.01%]).
In addition, the minimum provided in this table is the greatest percent decrease in the valuation of bitcoin for a single day, and respectively, the maximum is the greatest percent increase in the valuation of bitcoin for a single day. While the cummulative return is the total percentage return over the full period of the investment.
Finally, the value at risk is approximately 6.25% which means that on a daily time frame, there is a probability of \(5\%\) that one could lose 6.25% on the daily, this is important information for trading and risks could be taken to mitigate the risks of a 6.25% drawback in one’s portfolio through hedging or other means.
There two figure presented above is a representation of the returns plotted against time which denotes the returns for each of the change in price valuation on the daily basis. In addition, the histogram represents the frequency for the returns within a .5% interval, and based on the graph there are some outlier in the distribution but it is mostly centered around the mean and it has a gaussian shape meaning that the returns are normally distributed. This can be tested using a form of normality test but will not given the visual shape of the data.
This data was simulated using the mathematical equation: \[\delta S_t=S_{(t-1)}\cdot(\mu\delta t + \sigma \delta W_t)\] Where in this case \(\delta t = 1\) because the daily variance and the daily mean are being used rather than their annualized counter-parts.
| BullMarket (N = 1,068) | |
|---|---|
| Mean (sd) | 0.49 ± 3.72 |
| median (Q1, Q3) | 0.27 (-0.78, 1.83) |
| min | -21.14486 |
| max | 25.24717 |
| Cumulative Return | 520.3815 |
| Normally Distributed VaR | -5.6378883280451 |
| Non-Normally Distributed VaR | -5.35477918110908 |
| BearMarket (N = 894) | |
|---|---|
| Mean (sd) | 0.01 ± 4.18 |
| median (Q1, Q3) | 0.10 (-1.66, 1.63) |
| min | -37.16954 |
| max | 18.18776 |
| Cumulative Return | 8.210591 |
| Normally Distributed VaR | -6.87117808582232 |
| Non-Normally Distributed VaR | -6.45995118699981 |
| FirstMinorBearMarket (N = 363) | |
|---|---|
| Mean (sd) | -0.40 ± 4.39 |
| median (Q1, Q3) | -0.01 (-2.38, 1.46) |
| min | -16.8548 |
| max | 14.78049 |
| Cumulative Return | -143.6408 |
| Normally Distributed VaR | -7.61358726528272 |
| Non-Normally Distributed VaR | -8.58836887321155 |
| FirstMinorBullMarket (N = 194) | |
|---|---|
| Mean (sd) | 0.78 ± 3.46 |
| median (Q1, Q3) | 0.32 (-0.63, 1.91) |
| min | -8.831367 |
| max | 17.35601 |
| Cumulative Return | 150.7003 |
| Normally Distributed VaR | -4.91756452906168 |
| Non-Normally Distributed VaR | -4.72327966403773 |
| SecondMinorBearMarket (N = 175) | |
|---|---|
| Mean (sd) | -0.25 ± 3.85 |
| median (Q1, Q3) | -0.19 (-2.01, 1.35) |
| min | -14.08568 |
| max | 15.57634 |
| Cumulative Return | -44.38735 |
| Normally Distributed VaR | -6.58712229805485 |
| Non-Normally Distributed VaR | -6.31733229205686 |
| SecondMinorBearMarket (N = 60) | |
|---|---|
| Mean (sd) | 0.70 ± 2.71 |
| median (Q1, Q3) | 0.12 (-0.96, 1.96) |
| min | -4.211557 |
| max | 9.581901 |
| Cumulative Return | 41.95282 |
| Normally Distributed VaR | -3.75585094901244 |
| Non-Normally Distributed VaR | -2.981929208615 |
| SecondMinorBearMarket (N = 60) | |
|---|---|
| Mean (sd) | 0.70 ± 2.71 |
| median (Q1, Q3) | 0.12 (-0.96, 1.96) |
| min | -4.211557 |
| max | 9.581901 |
| Cumulative Return | 41.95282 |
| Normally Distributed VaR | -3.75585094901244 |
| Non-Normally Distributed VaR | -2.981929208615 |
| SeconMinorBullMarket (N = 2,053) | |
|---|---|
| Mean (sd) | 2,052; 0.26 ± 3.86 |
| median (Q1, Q3) | 2,052; 0.20 (-1.21, 1.79) |
| min | -21.14486 |
| max | 25.24717 |
| Cumulative Return | 528.0639 |
| Normally Distributed VaR | -6.09523722010413 |
| Non-Normally Distributed VaR | -6.04396999141942 |
For the tables presented above, the shorter term trend changes which occur within the Major Bear market are labeled as “minor.” These minor changes are time frames where a local high was either surpased denoting a bullish change in price valuation or a local low was broken denoting a bearish change in price valuation. The returns of Bitcoin can be compared for each trending period using these tables; in addition, these tables exemplifies the importance of timing an investment correctly and investing in the bottom is not necessary so long as an investment is initialized early in the bullish trend.
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 30.8 2991.6 9649.9 41609.1 32160.4 1593239.7| Price | |
|---|---|
| 5% | 494.7718 |
| 10% | 971.8468 |
| 15% | 1480.1073 |
| 20% | 2111.7316 |
| 25% | 2991.5725 |
| 30% | 3773.8742 |
| 35% | 4888.8804 |
| 40% | 6088.6554 |
| 45% | 7744.6184 |
| 50% | 9649.8751 |
| 55% | 12371.7051 |
| 60% | 15580.4536 |
| 65% | 20077.5926 |
| 70% | 26069.7368 |
| 75% | 32160.3775 |
| 80% | 43399.7529 |
| 85% | 61068.0143 |
| 90% | 92737.5045 |
Based on the simulation of 1000 prices using random walk theory, a summary of the statistics can be created in which a cummulative probability of the randomized projections of the price is obtained. This simply means that 95% of the data is \(\le\) to the number given in the simulation.
The distribution of the simulated data can be seen in the histogram given as a exponential distribution. The reason for this is because there is a lower limit occuring in the data.